Why does my rig make my arm skinnier?
Be sure to apply (CTRL+a) scaling, before you assign your bonen armature to the mesh. This is a common fault, which people forget todo when they start with rigging. 

If after you still have such a problem, it might be caused by the weight paint of the bone, the more easy way to solve that might be (depending on mesh) to add some helper bones to keep things in place how you want it.